I'm trying to understand how a prediction from a multinomial logit model compares to a prediction from many logit models. A few videos and wikipedia have lead to me to believe that the categorical prediction from a multinomial model is equivalent to the prediction of the logit model for which the prediction is the highest.
As an example, I have 4 specific (ranked) outcomes: 1,2,3,4. Each item in the dataset has continuous independent variables and 4 different indicator variables to indicate if the response is 1,2,3, or 4. I then run a logistic regression for each response and obtain 4 sets of fitted betas. Each of these sets of coefficients are applied to a new X that I want to predict. I can then make four predictions of the dependant variable y, one for each outcome. Given the link function, this will be transformed into a probability of how likely this new X is to fall into each of the 4 outcomes.
If, for example, the single logit model predicts that outcome 3 will occur with a 29% probability, and that this is the higher than outcomes 1,2, or 4, is this the prediction that the multinomial logit model would provide as well?